Premise Selection for Mathematics by Corpus Analysis and Kernel Methods

Smart premise selection is essential when using automated reasoning as a tool for large-theory formal proof development. A good method for premise selection in complex mathematical libraries is the application of machine learning to large corpora of proofs. This work develops learning-based premise selection in two ways. First, a newly available minimal dependency analysis of existing high-level formal mathematical proofs is used to build a large knowledge base of proof dependencies, providing precise data for ATP-based re-verification and for training premise selection algorithms. Second, a new machine learning algorithm for premise selection based on kernel methods is proposed and implemented. To evaluate the impact of both techniques, a benchmark consisting of 2078 large-theory mathematical problems is constructed,extending the older MPTP Challenge benchmark. The combined effect of the techniques results in a 50% improvement on the benchmark over the Vampire/SInE state-of-the-art system for automated reasoning in large theories.Comment: 26 page

Zinc is a transmembrane agonist that induces platelet activation in a tyrosine phosphorylation-dependent manner

Following platelet adhesion and primary activation at sites of vascular injury, secondary platelet activation is induced by soluble platelet agonists, such as ADP, ATP, thrombin and thromboxane. Zinc ions are also released from platelets and damaged cells and have been shown to act as a platelet agonist. However, the mechanism of zinc-induced platelet activation is not well understood. Here we show that exogenous zinc gains access to the platelet cytosol and induces full platelet aggregation that is dependent on platelet protein tyrosine phosphorylation, PKC and integrin αIIbβ3 activity and is mediated by granule release and secondary signalling. ZnSO4 increased the binding affinity of GpVI, but not integrin α2β1. Low concentrations of ZnSO4 potentiated platelet aggregation by collagen-related peptide (CRP-XL), thrombin and adrenaline. Chelation of intracellular zinc reduced platelet aggregation induced by a number of different agonists, inhibited zinc-induced tyrosine phosphorylation and inhibited platelet activation in whole blood under physiologically relevant flow conditions. Our data are consistent with a transmembrane signalling role for zinc in platelet activation during thrombus formation

Formalising Mathematics in Simple Type Theory

Despite the considerable interest in new dependent type theories, simple type theory (which dates from 1940) is sufficient to formalise serious topics in mathematics. This point is seen by examining formal proofs of a theorem about stereographic projections. A formalisation using the HOL Light proof assistant is contrasted with one using Isabelle/HOL. Harrison's technique for formalising Euclidean spaces is contrasted with an approach using Isabelle/HOL's axiomatic type classes. However, every formal system can be outgrown, and mathematics should be formalised with a view that it will eventually migrate to a new formalism

Université Catholique de Louvain Groups

Summary. Notions of group and abelian group are introduced. The power of an element of a group, order of group and order of an element of a group are defined. Basic theorems concerning those notions are presented